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November 2012
Brie Finegold summarizes blogs on math and a nonreversing mirror invention and math for fiction writers: "Math Professor Invents NonReversing Mirror," by James Anderson. Techfragments, 18 November 2012. Take a photograph of yourself in the mirror and that mole just above your right eye is now above your left. But putting a curve in the surface of the mirror changes everything. Try looking at your reflection in either side of a spoon. When you look at the concave side, the top and bottom of the image are reversed. Why is it the top and bottom that are reversed and not the northwest and southeast corners for instance? Differential topologist and computer scientist Dr. Andrew Hicks would know the answer. In 2009, Hicks, who is from Drexel University in Philadelphia, PA, invented a nonreversing mirror by curving the surface back and forth in just the right manner. More recently, Hicks patented a mirror that eliminates the blind spot when driving. While he has yet to come up with any practical applications for his nonreversing mirror, it has attracted the attention of New York artist Robin Cameron and is currently being displayed as part of Cameron's art exhibition. "Why Fiction Writers Should Learn Math," by Alexander Nazaryan. PageTurner Blog, The New Yorker, 2 November 2012. As mathematicians, we know that strong language skills are a must for success in our profession. But this blog post focuses on why even novelists should strive to understand mathematics. Blogger Nazaryan emphasizes the patternmaking skills involved in both disciplines. His quotes from famous mathematicians like G.H. Hardy and Terence Tao mostly focus on how mathematicians view themselves as artists in some respects. There is also mention of the courage necessary to create original ideas in both fields. Several examples of writers who gathered inspiration from mathematics are given, including David Foster Wallace, the author of Infinite Jest. The central theme revolves around finding the balance between structure and freeflowing associations when creating original work. This post provoked a fair amount of heated commentary from readers which is itself entertaining to read. In particular, some people balked at the idea that creativity is required for mathematics in the same way that it is for poetry, for instance. In any case, Nazaryan bemoans the lack of courage in modern fiction writing and implies that having a mathematical bent allows one to better see structure and therefore choose when to adhere to it and when to break it.  Brie Finegold "Forecasting cancer," by Katharine Gammon. Nature, 22 November 2012, pages S6667. Cancer results when genetic mutations release certain cells from the nonproliferation agreements at the core of the cellular society that is the body. Like invasive species, these renegade cells disrupt the body's harmonious ecology, thriving at the expense of healthy cells. Reproducing rapidly, they continue to mutate and adapt to the changing internal environmentmaking them hard to pin down and hard to treat, like, er, a writer who keeps changing metaphors, and maybe also has the flu. In any case, approaching cancer from an evolutionary and gametheoretic perspective not only makes sensein the hands of mathematicians, who thrive in environments where models from one domain can be applied to anotherit is yielding surprising insights into how these invasive cells grow, and how best to fight them. For example, mathematical biologists at the Moffitt Cancer Center in Tampa, FL, have used models incorporating the microenvironment of tumor cells, as well as more traditional histological measures, to obtain more accurate predictions about tumor growth. The evolutionary perspective also sheds light on why "targeted" cancer treatments often turn out to be less effective than doctors and researchers hope: targeting particular mutations creates an environmental pressure that cancer cells can adapt to, resulting in resistance rather than remission. "Cancer comes up, and we whack it and do another treatment," says Moffitt Center oncologist Robert Gatenby, "We need to plan strategically how we treat patients, so that as we give one therapy, we’re producing an adaptive response to be anticipated with our second therapy. Instead of whac[k]amole, we need to be playing chess." Gametheoretic modeling is now guiding treatments aimed at taking advantage of cancer's shortsightednessit can only look one step aheadto maneuver breast tumors into drugsusceptible states. Biostatistician Franziska Michor is using mathematical models to predict the optimal dosing scheme for lung cancer, and oncologist Larry Norton has used insights regarding tumor growth curves to obtain better breast cancer treatment results by delivering the same dose of chemotherapy in a shorter time, giving young tumors less time to regrow between treatments. As Norton says, "The biological sciences have become antimaths, and it’s just not working. Researchers see lots and lots of facts, but they’re not connecting them together,” he says. The missing ingredient is mathematicians, who thankfully are making the pilgrimmage to biology.  Ben Polletta "Math is behind cozy penguin huddles," by Stephanie Pappas. NBC News, 16 November 2012. Inspired by the documentary, "The March of the Penguins," a team of researchers has identified and quantified the huddling behavior of Emperor Penguins, in which the penguins on the outer perimeter move to downwind boundaries for shelter. The mathematical model takes into account varying wind strength and turbulence around the huddle, unpredictable movement of the individuals and the temperature redistribution as huddle shape changes. While "each individual penguin in the huddle seeks only to reduce its own heat loss," the result in the end is that "all penguins [have] approximately equal access to the warmth of the huddle." One of the researchers, Francois Blanchette, an applied mathematician at UC Merced who focuses on fluid dynamics, tells NBC news that "his group may also investigate how to adapt the model to describe other biological organisms, such as certain bacteria, that move as a group in response to an outside stimulus like food or the presence of a toxin. Eventually, concepts from the model may guide the design of swarming robots that shelter each other in harsh conditions." In the meantime, he hopes that the research may inspire people to see mathematics from another and positive perspective: He says, "Nearly everybody seems to love penguins and not enough people love math. If we use math to study penguins we could potentially teach more people to love math too!" The original paper is "Modeling Huddling Penguins," by Aaron Waters, François Blanchette, Arnold D. Kim. PLOS One, 16 November 2012.  Annette Emerson "Brainzapping Kinect game boosts mathematical skills", by Sandrine Ceurstemont. New Scientist TV, 16 November 2012. In this video, a New Scientist editor tries out a Kinect video game in a cap that could help develop mathematical skills. The cap delivers electrical stimulation to the right parietal cortex of the brain, a region associated with numerical understanding. According to the article, the research team working with this video game "has shown that brain stimulation while doing computerbased mathematics exercises helped maintain better mathematical skills in adults even six months later."  Allyn Jackson "Our brain can do unconscious mathematics," by Douglas Heaven. New Scientist, 13 November 2012. This short article describes a psychological experiment purporting to demonstrate that some arithmetic calculations are done subconsciously. The experiment used a technique called "continuous flash suppression", which the article describes this way: "The technique works by presenting a volunteer's left eye with a stimulusa mathematical sum, sayfor a short period of time, while bombarding the right eye with rapidly changing colourful shapes. The volunteer's awareness is dominated by what the right eye sees, so they remain unconscious of what is presented to the left eye." After this, both eyes are presented with a number, which the experimental subjects were to say aloud. When the number equalled the sum, the subjects were faster to say the number than when the number was different from the sum. This suggests "that they had subconsciously worked out the answer, and primed themselves with that number," the article claims. The article tries to bolster this claim by describing outcomes from analogous experiments carried out with words.  Allyn Jackson "Daniel Tammet: 'Maths is as rich, inspiring and human as literature'," by Adam Feinstein. The Guardian, 10 November 2012. The Observer's 'Meet the author' series interviews Daniel Tammet, a bestselling mathematical savant who has Asperger's syndrome. Tammet's newest book, Thinking in Numbers: How Maths Illuminates Our Lives explores mathematics as the science of imagination. Tammet is not a professional mathematician but mathematics play a big part in his storytelling. As a kid Tammet had difficulties understanding social interactions, isolating him in school. However it wasn't until 2004 when he achieved the European record for reciting pi that he was diagnosed with Asperger's. Tammet's first book Born on a Blue Day found large and royal readership and he now lives in Paris and works as a fulltime writer. He is currently working on a novel based on the 1972 FischerSpassky chess match in Reykjavik.  Baldur Hedinsson "Math takes center stage as experts predict storms, elections and more." Oregon Live, 10 November 2012. The correct predictions of Hurricane Sandy's assault on New Jersey and President Obama's electoral victory in late October and early November were major wins for the statisticians behind them. In both cases, statisticians used large computing power to run models based on mathematical formulas and arrived at specific probabilities that certain events would occur. In the case of the election, blogger Nate Silver used a model that weighted recent polling data based on previous accuracy and performed over 40,000 simultaneous computer simulations. His method correctly predicted the electoral wins in all 50 states, including Florida's initial tie that eventually went to President Obama. Computer modeling already is in wide use commercially and within the government, from nuclear weapons blast effects to Pringles can designs. Statisticians cite these recent victories as indications that the advancing field will produce additional powerful predictions, such as the location of foodpoisoning outbreaks, in the future. (Image: Predicting Storm Surge, from the Mathematical Moments program.)  Lisa DeKeukelaere Math expert helped end WWII," by Jason Brown. The Chronicle Herald, 9 November 2012. To mark Remembrance Day in Canada, Dalhousie University professor of mathematics Jason Brown tells the story of how 24yearold mathematician William Tutte "changed the course of the [Second World War] and likely saved millions of lives." Tutte was a chemistry graduate student at Cambridge University in 1941 when he was sent to assist in the secret codebreaking operations at Bletchley Park. "Some messages from the German high command had been intercepted by British intelligence, to no effect," writes Brown. "A single mistake by a German coding operator led to the sending of two very similar messages with the identical key." After four months and little progress by the research group, the problem was given to Tutte who "noticed patterns in groups of 41 characters, patterns that were likely not random, and from this he inferred that the machine must have had a wheel with 41 teeth." Further work on the problem "led Tutte and his colleagues to determine completely the structure of the internal workings of the Lorenz machine." After the war, Tutte received his doctorate in mathematics at Cambridge and went on to teach and do research at the University of Toronto, followed by the University of Waterloo. He died in 2002 at age 84.  Claudia Clark "John Mighton's The Little Years offers inspiration," by Bruce DeMara. Toronto Star, 9 November 2012. Playwright and mathematician John Mighton did not think he was talented at either writing or math, but inspiration led him to success in both fields, and his experience of selfdoubt and inspiration formed the foundation for his play The Little Years, now showing in Toronto. After receiving low marks in creative writing in college, Mighton found inspiration in the work of Sylvia Plath and attempted to teach himself to write "by sheer determination" and by imitating works he loved. After failing his college calculus class, Mighton turned away from math until an advertisement for a high school math teacher led him back to a field he once found fascinating, where he developed a deeper understanding of the concepts by explaining them to others and eventually earned a doctorate. Mighton links the storyline of The Little Years, the tale of a woman who struggles to find herself despite perceived failures, to the goals of the Junior Undiscovered Math Prodigies program, which he founded. (Photo of John Mighton by Chris Chapman, courtesy of JUMP.)  Lisa DeKeukelaere "Mathematical proof reveals magic of Ramanujan's genius," by Jacob Aron. New Scientist, 8 November 2012. In 1920, in his last letter to his mentor and collaborator G. H. Hardy, Srinivasa Ramanujan wrote down a handful of functions for which he claimed a number of mysterious propertiesincluding detailing their "orders" and claiming that they would approximate theta functions at the roots of unity. As was his habit, he made these claims without explanation or proof, and in the next several decades other mathematicians worked to uncover the rigorous foundations of his remarkable intuitions. But they remained unable to give a formal definition of these functions, called mock theta functions, or to generate many more examples than those Ramanujan himself provided. It was not until 2001 that Sander Zwegers gave a formal definition of mock modular forms, encompassing mock theta functions and allowing for an infinite number of these functions to be constructed. Zwegers found that Ramanujan's notion of order corresponds to "the conductor of the Nebentypus character of the weight 1/2 harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections". Yet several of Ramanujan's claims about these functions remained poorly understood  including the claim that these functions would approximate modular forms at the roots of unity. Now, on Ramanujan's 125th birthday, number theorist Ken Ono, an expert in mock modular forms, has finally figured out just what Ramanujan meant by his claim. He's computed the values of a number of mock and real modular forms at 1 (one of the square roots of unity), and found that, while the modular forms diverge to infinity, the mock modular forms diverge to minus infinity. But the sum of the mock and real values limits on the small, finite value of fourso the mock modular forms really do approximate their authentic twins at this value. The perplexing thing is that the technology Ono used to make these calculations was developed by him in 2006so it seems unlikely that Ramanujan used the same method to arrive at his claims. How the Indian virtuoso knew where the mock theta functions would approximate the real thing will remain as mysterious as the remarkable insight that allowed him to invent the functionsand so much other new mathematicsin the first place, and to intuit some of their deepest properties. (Photo: Oberwolfach Photo Collection, Creative Commons AttributionShare Alike 2.0 Germany license. ) [Editor's note: The December issue of Notices of the American Mathematical Society commemorates Ramanujan's 125^{th} birthday this month with a series of articles on Ramanujan's life and ideas: "Srinivasa Ramanujan: Going Strong at 125, Part I," edited by Krishnaswami Alladi, with contributions by Alladi, George E. Andrews, Bruce C. Berndt, and Jonathan M. Borwein. the series will continue in the January issue "Part II" with contributions by Ken Ono, Kannan Soundararajan, Robert Vaughan and S. Ole Warnaar.]  Ben Polletta "Africa's counting house," by Leigh Phillips. Nature, 8 November 2012, pages 1768. The counting house is the African Institute for Mathematical Sciences in South Africa. It was founded by cosmologist Neil Turok, who grew up in South Africa, as a way of helping Africa develop. He says that "There is nothing more costeffective for development than mathematics." Students come from Africa while teachersincluding two Fields Medalistscome from all over the globe. The tuitionfree program is intensive, with three weeks of courses and a sevenweek research project. Turok has opened other institutes in Senegal and Ghana, and one is planned in Ethiopia. He says that his goal is to have 15 institutes throughout Africa.  Mike Breen Articles on math causing pain: "High Anxiety: How Worrying About Math Hurts Your Brain," by Olivia B. Waxman. Time, 8 November 2012; "Math Can Be Truly Painful, Brain Study Shows," by Jeremy Berlin. National Geographic News, 8 November 2012; "Math Problems Can Be Physically Painful," by Ian Steadman. Wired, 5 November 2012; "Fear of math can cause real pain," by Megan Gannon. CBS News, 2 November 2012; "Brain Scan Shows That Thinking About Math Is As Painful As A Hot Stove Burn, If You're Anxious," by Rebecca Boyle. Popular Science, 1 November 2012. According to a study published on October 31 in the online journal, PLOS One, people who are math anxious experience physical pain when they anticipate solving math problems. The study, "When Math Hurts: Math Anxiety Predicts Pain Network Activation in Anticipation of Doing Math," was conducted by Ian M. Lyons and Sian L. Bielock, psychology professors at the University of Chicago. Beilock and Lyons used a functional MRI to take brain scans of 28 people (14 with high math anxiety and 14 with low math anxiety) while they anticipated performing a math task and then while performing the task. Out of the 28 individuals, only the 14 individuals with high math anxiety showed increased brain activity in the dorsoposterior insula (INSp), "the fold of tissue in the brain that is activated when a person experiences physical pain," when they knew they were going to have to solve a math problem. "Interestingly," writes Waxman, "these pain areas were more intensely activated when the mathphobics were anticipating an upcoming mathrelated task, and not while they were actually trying to solve a math problem." It is unlikely that this response to math is rooted in evolutionary survival strategies. Instead, Waxman notes, to Lyons and Beilock "the data suggest that even nonthreatening, physically benign circumstances can tap into the anxiety and pain response—if we let them." Articles about this study appeared in a number of other media outlets, including Popular Science, Wired, National Geographic Daily News, and CBS News.  Claudia Clark "MIT algorithm predicts Twitter trending topics up to five hours in advance," by Ian Steadman. Wired, 2 November 12. MIT professor Devavrat Shah and his student, Stanislav Nikolov, have developed an algorithm that can predict trending topics on Twitter an average of an hour and a half before they appear. Their algorithm can predict with "95 percent accuracy" which words, phrases or hashtags will end up trending, even up to four or five hours in advance. They do this by "training" the algorithm, teaching it how to work out the patterns that will indicate what's likely to become a popular topic. This discovery might have commercial implications for Twitter since the company might be able to use the algorithm to detect popular topics and charge for ads linked to those topics.  Baldur Hedinsson "The Natural Beauty of Math," by Dana Mackenzie. Smithsonian Magazine, November 2012. While beauty in the art world may be a subjective concept, the author evaluates the "mathematical beauty" of a theorem based on three specific tenets: 1) it is surprising; 2) it is simple; and 3) it contains many layers of meaning. For example, Roger A. Johnson's 1916 theoremthat for three identical circles that intersect at a single point, the size of the circle needed to enclose the three remaining intersection points is the same as the original circlesoffers a "taste of beauty," meeting two of the three qualifications (Mackenzie feels that it falls short of tenet 3). German mathematician Stefan Friedl has contended that Grigory Perelman's Geometrization Theorem, proven in 2003, is a prime example of mathematical beauty. The theorem involves the classification of threedimensional topological spaces and the curvature of space, a departure from the flat, Euclidean universe that we often picture.  Lisa DeKeukeleare "The Science of Sticky Spheres," by Brian Hayes. American Scientist, NovemberDecember 2012, pages 442449. A group of two spheres can have at most one contact point, a group of three spheres can have at most three contact points, but what about a group of 12? Hayes explores the problem of determining the maximum number of contact points, and the number of unique configurations that achieve this maximum. Spherepacking conjectures date back to Kepler's declaration that a grocer's pyramid of oranges is the tightest possible combination of spheres, followed by Newton's theory that a sphere can touch a maximum of 12 other spheres. More recently, Harvard and Yale researchers mapped out the maximum number of contact points for groups of up to 11 spheres by using matrices of 0's and 1's (denoting the absence or presence of contact between two spheres) to check all possible configurations for a given group size. Using numerical and geometric methods, the researchers determined which matrices corresponded to geometrically feasible configurations, and subsequently identified each of the maximumcontact solutions. The researchers' interest stemmed from questions regarding real material spheres, such as during crystallization and selfassembly of nanostructures. (Image courtesy of Brian Hayes.)  Lisa DeKeukelaere

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